In a polynomial fraction #f(x) = (p_n(x))/(p_m(x))# we have:

#1)# vertical asymptotes for #x_v# such that #p_m(x_v)=0#

#2)# horizontal asymptotes when #n le m#

#3)# slant asymptotes when #n = m + 1#

In the present case we have #x_v = -2# and #n = m+1# with #n = 2# and #m = 1#

Slant asymptotes are obtained considering #(p_n(x))/(p_{n-1}(x))
approx y = a+b x# for large values of #abs(x)#

In the present case we have

#(p_n(x))/(p_{n-1}(x)) = (2x^2-3x+4)/(x + 2) #

#p_n(x) = p_{n-1}(x)(a x + b) + r_{n-2}(x)#

#(p_n(x))/(p_{n-1}(x)) = (2x^2-3x+4) =(x+2) (a x +b)+c#

equating we have

#{
(4 - 2 b - c=0), (-3 - 2 a - b=0), (2 - a=0)
:}#

Solving for #a,b,c# we have #a = 2,b=-7,c=18#

so the slant asymptote reads

#y = 2x-7 #